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Gromov-Witten invariants on Grassmannians. (English) Zbl 1063.53090
The main result of the paper under review is: any three-point genus zero Gromov-Witten invariant on a Grassmannian \(X\) is equal to a classical intersection number on a homogeneous space \(Y\) of the same Lie type. The statement was proven when \(X\) is a type \(A\) Grassmannian, and, in types \(B\), \(C\), and \(D\), when \(X\) is the Lagrangian or orthogonal Grassmannian parametrizing maximal isotropic subspaces in a complex vector space equipped with a non-degenerate skew-symmetric or symmetric form. Their key identity for Gromov-Witten invariants is based on an explicit bijection between the set of rational maps counted by a Gromov-Witten invariant and the set of points in the intersection of three Schubert varieties in the homogeneneous space \(Y\). It should be noted that the proof of their result does not use moduli spaces of maps and only uses basic algebraic geometry. In types \(B\), \(C\), and \(D\), their result may be used to give new proofs of the structure theorems for the quantum cohomology of Lagrangian and orthogonal Grassmannians obtained by A. Kresch and H. Tamvakis in [J. Algebr. Geom. 12, No. 4, 777–810 (2003; Zbl 1051.53070) and Compos. Math. 140, No. 2, 482–500 (2004; Zbl 1077.14083)]. Their methods can also be used to prove a quantum Pieri rule for the quantum cohomology of sub-maximal isotropic Grassmannians.

MSC:
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14N15 Classical problems, Schubert calculus
05E15 Combinatorial aspects of groups and algebras (MSC2010)
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References:
[1] L. Bégin, C. Cummins, and P. Mathieu, Generating-function method for fusion rules, J. Math. Phys. 41 (2000), no. 11, 7640 – 7674. · Zbl 1050.17020
[2] L. Bégin, A. N. Kirillov, P. Mathieu, and M. A. Walton, Berenstein-Zelevinsky triangles, elementary couplings, and fusion rules, Lett. Math. Phys. 28 (1993), no. 4, 257 – 268. · Zbl 0811.17032
[3] L. Bégin, P. Mathieu, and M. A. Walton, \Hat \?\?(3)_{\?} fusion coefficients, Modern Phys. Lett. A 7 (1992), no. 35, 3255 – 3265. · Zbl 1021.81530
[4] Nantel Bergeron and Frank Sottile, Schubert polynomials, the Bruhat order, and the geometry of flag manifolds, Duke Math. J. 95 (1998), no. 2, 373 – 423. · Zbl 0939.05084
[5] Aaron Bertram, Quantum Schubert calculus, Adv. Math. 128 (1997), no. 2, 289 – 305. · Zbl 0945.14031
[6] Aaron Bertram, Ionuţ Ciocan-Fontanine, and William Fulton, Quantum multiplication of Schur polynomials, J. Algebra 219 (1999), no. 2, 728 – 746. · Zbl 0936.05086
[7] A. S. Buch : Quantum cohomology of Grassmannians, Compositio Math., to appear. · Zbl 1050.14053
[8] A. S. Buch : A direct proof of the quantum version of Monk’s formula, Proc. Amer. Math. Soc., to appear. · Zbl 1012.14018
[9] A. S. Buch, A. Kresch, and H. Tamvakis : Grassmannians, two-step flags, and puzzles, in preparation.
[10] A. S. Buch, A. Kresch, and H. Tamvakis : Quantum Pieri rules for isotropic Grassmannians, in preparation. · Zbl 1053.05121
[11] Sergey Fomin and Anatol N. Kirillov, Quadratic algebras, Dunkl elements, and Schubert calculus, Advances in geometry, Progr. Math., vol. 172, Birkhäuser Boston, Boston, MA, 1999, pp. 147 – 182. · Zbl 0940.05070
[12] William Fulton, Young tableaux, London Mathematical Society Student Texts, vol. 35, Cambridge University Press, Cambridge, 1997. With applications to representation theory and geometry. · Zbl 0878.14034
[13] W. Fulton and R. Pandharipande, Notes on stable maps and quantum cohomology, Algebraic geometry — Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 45 – 96. · Zbl 0898.14018
[14] Howard Hiller and Brian Boe, Pieri formula for \?\?_{2\?+1}/\?_{\?} and \?\?_{\?}/\?_{\?}, Adv. in Math. 62 (1986), no. 1, 49 – 67. · Zbl 0611.14036
[15] A. Knutson : Private communication.
[16] A. Knutson, T. Tao and C. Woodward : The honeycomb model of \(GL(n)\) tensor products II: Puzzles determine facets of the Littlewood-Richardson cone, J. Amer. Math. Soc., to appear. · Zbl 1043.05111
[17] A. Kresch and H. Tamvakis : Quantum cohomology of the Lagrangian Grassmannian, J. Algebraic Geom., to appear. · Zbl 1051.53070
[18] A. Kresch and H. Tamvakis : Quantum cohomology of orthogonal Grassmannians, Compositio Math., to appear. · Zbl 1077.14083
[19] Piotr Pragacz, Algebro-geometric applications of Schur \?- and \?-polynomials, Topics in invariant theory (Paris, 1989/1990) Lecture Notes in Math., vol. 1478, Springer, Berlin, 1991, pp. 130 – 191. · Zbl 0783.14031
[20] P. Pragacz and J. Ratajski, Formulas for Lagrangian and orthogonal degeneracy loci; \?-polynomial approach, Compositio Math. 107 (1997), no. 1, 11 – 87. · Zbl 0916.14026
[21] Bernd Siebert and Gang Tian, On quantum cohomology rings of Fano manifolds and a formula of Vafa and Intriligator, Asian J. Math. 1 (1997), no. 4, 679 – 695. · Zbl 0974.14040
[22] Frank Sottile, Pieri’s formula for flag manifolds and Schubert polynomials, Ann. Inst. Fourier (Grenoble) 46 (1996), no. 1, 89 – 110 (English, with English and French summaries). · Zbl 0837.14041
[23] G. Tudose : On the combinatorics of \(sl(n)\)-fusion coefficients, preprint (2001).
[24] A. Yong : Degree bounds in quantum Schubert calculus, Proc. Amer. Math. Soc., to appear. · Zbl 1064.14066
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